On the Affine Homogeneity of Algebraic Hypersurfaces Arising from Goernstein Algebras

Abstract

To every Gorenstein algebra $A$ of finite vector space dimension greater than 1 over a field $\mathbb{F}$ of characteristic zero, and a linear projection $\pi$ on its maximal ideal $\mathfrak{m}$ with range equal to the annihilator $\operatorname{Ann}(\mathfrak{m})$ of $\mathfrak{m}$, one can associate a certain algebraic hypersurface $S_{\pi} \subset \mathfrak{m}$. Such hypersurfaces possess remarkable properties. They can be used, for instance, to help decide whether two given Gorenstein algebras are isomorphic, which for $\mathbb{F} = \mathbb{C}$ leads to interesting consequences in singularity theory. Also, for $\mathbb{F} = \mathbb{R}$ such hypersurfaces naturally arise in CR-geometry. Applications of these hypersurfaces to problems in algebra and geometry are particularly striking when the hypersurfaces are affine homogeneous. In the present paper we establish a criterion for the affine homogeneity of $S_{\pi}$ . This criterion requires the automorphism group $\operatorname{Aut}(\mathfrak{m})$ of $\mathfrak{m}$ to act transitively on the set of hyperplanes in m complementary to $\operatorname{Ann}(\mathfrak{m})$. As a consequence of this result we obtain the affine homogeneity of $S_{\pi}$ under the assumption that the algebra $A$ is graded.